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Electric Field To Potential

  1. Mar 21, 2017 #1
    1. The problem statement, all variables and given/known data
    I have a potential Value like ##V=V(x,y,z)+C##
    I found ##\vec E## using partial derivative, like ##\vec E=((-∂V/∂x)i+(-∂V/∂y)j+(-∂V/∂z) k)##
    Theres two position vectors,
    ##\vec r_{a}=2i##
    ##\vec r_{b}=j+k##
    We need to find ##V_{ba}=?##
    2. Relevant equations
    ##V_b-V_a=-\int_{r_{a}}^{r_{b}} \vec E⋅d\vec r##
    ##V_r=V(x_i,y_i,z_i)## where ##r=(x_i,y_i,z_i)##

    3. The attempt at a solution
    Ok I found E but since we are taking partial derivative the constant term disappeared.

    I can find from ##V_b=V(0,1,1)## and ##V_a=V(2,0,0)## and the difference will be ##V_b-V_a=V_{ba}##

    But If ı try to do this from ##V_b-V_a=-\int_{r_{a}}^{r_{b}} \vec E⋅d\vec r## using this.How can I approach the question.##\vec E## is a function of ##x,y,z## but we need a function of ##\vec r##

    I mean the confusing part is,
    ##V_b-V_a=-\int_{r_{a}}^{r_{b}} ((-∂V/∂x)i+(-∂V/∂y)j+(-∂V/∂z) k)⋅d\vec r##

    How can I take integral in this case ?

    I ll do ##\vec E⋅\vec r_a-\vec E⋅\vec r_b## ??

    And is my approach or answer is true..? , Is a constant term here makes a diffference ?
     
  2. jcsd
  3. Mar 21, 2017 #2

    BvU

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    Hi,

    If you have an expression for V, why go the long way via ##\vec E## and integration if you can simply take ##V_b-V_a## ?
     
  4. Mar 21, 2017 #3
    Just curiosity :)
     
  5. Mar 21, 2017 #4

    BvU

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    Write ##d\vec r## as ##dx \, {\bf\hat\imath} + dy \, {\bf\hat\jmath}+ dz\, {\hat k}## and write out the dot product to give you three terms of the integrand in three integrals...
     
  6. Mar 21, 2017 #5
    so ##V_b-V_a=-\int_{x=0}^{2}\int_{y=0}^{1}\int_{z=0}^{1} ((-∂V/∂x)+(-∂V/∂y)+(-∂V/∂z))dxdydz##
     
  7. Mar 21, 2017 #6
    or maybe
    ##V_b-V_a=-\int_{x=0}^{2}((-∂V/∂x)dx+\int_{y=0}^{1}(-∂V/∂y)dy+\int_{z=0}^{1}(-∂V/∂z))dz##
     
  8. Mar 21, 2017 #7
    Are these two integral same ?
     
  9. Mar 21, 2017 #8

    BvU

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    Try an example, e.g. ##V(x, y, z) = 2x+3y+4z + 1199## :smile:
     
  10. Mar 21, 2017 #9

    BvU

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    No. dxdydz is a volume integral. That's not the idea...
     
  11. Mar 21, 2017 #10
    I see you are right
    But it must be the
    I did your example and it gave me the same thing.I understand the idea.But the constant dissapered which thats bothers.Or maybe it didnt.

    And thanks :smile::smile:
     
  12. Mar 21, 2017 #11
    I think in the calculating V difference, the constant has no affect.Of course it doesnt thats the logical thing...
     
  13. Mar 21, 2017 #12
    Ok thanks a lot again
     
  14. Mar 21, 2017 #13

    BvU

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    Correct: a potential is basically unnoticeable. Only potential differences bring about something that can be sensed.
     
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